Fatou, Julia, and escaping sets of conjugate holomorphic semigroups

Abstract

We define commutator of a holomorphic semigroup, and on the basis of this concept, we define conjugate semigroups of a holomorphic semigroup. We prove that the conjugate semigroup is nearly abelian if and only if the given holomorphic semigroup is nearly abelian. We also prove that image of each of Fatou, Julia, and escaping sets of a holomorphic semigroup under commutator (affine complex conjugating map) is equal respectively, to the Fatou, Julia, and escaping sets of the conjugate semigroup. Finally, we prove that every element of a nearly abelian holomorphic semigroup S can be written as the composition of an element from the set generated by the set of commutators !(S) and the composition of the certain powers of its generators..